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Mirrors > Home > ILE Home > Th. List > acexmidlemcase | Unicode version |
Description: Lemma for acexmid 5741. Here we divide the proof into cases (based
on the
disjunction implicit in an unordered pair, not the sort of case
elimination which relies on excluded middle).
The cases are (1) the choice function evaluated at equals , (2) the choice function evaluated at equals , and (3) the choice function evaluated at equals and the choice function evaluated at equals . Because of the way we represent the choice function , the choice function evaluated at is and the choice function evaluated at is . Other than the difference in notation these work just as and would if were a function as defined by df-fun 5095. Although it isn't exactly about the division into cases, it is also convenient for this lemma to also include the step that if the choice function evaluated at equals , then and likewise for . (Contributed by Jim Kingdon, 7-Aug-2019.) |
Ref | Expression |
---|---|
acexmidlem.a | |
acexmidlem.b | |
acexmidlem.c |
Ref | Expression |
---|---|
acexmidlemcase |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | acexmidlem.a | . . . . . . . . . . . . . 14 | |
2 | onsucelsucexmidlem 4414 | . . . . . . . . . . . . . 14 | |
3 | 1, 2 | eqeltri 2190 | . . . . . . . . . . . . 13 |
4 | prid1g 3597 | . . . . . . . . . . . . 13 | |
5 | 3, 4 | ax-mp 5 | . . . . . . . . . . . 12 |
6 | acexmidlem.c | . . . . . . . . . . . 12 | |
7 | 5, 6 | eleqtrri 2193 | . . . . . . . . . . 11 |
8 | eleq1 2180 | . . . . . . . . . . . . . . 15 | |
9 | 8 | anbi1d 460 | . . . . . . . . . . . . . 14 |
10 | 9 | rexbidv 2415 | . . . . . . . . . . . . 13 |
11 | 10 | reueqd 2613 | . . . . . . . . . . . 12 |
12 | 11 | rspcv 2759 | . . . . . . . . . . 11 |
13 | 7, 12 | ax-mp 5 | . . . . . . . . . 10 |
14 | riotacl 5712 | . . . . . . . . . 10 | |
15 | 13, 14 | syl 14 | . . . . . . . . 9 |
16 | elrabi 2810 | . . . . . . . . . 10 | |
17 | 16, 1 | eleq2s 2212 | . . . . . . . . 9 |
18 | elpri 3520 | . . . . . . . . 9 | |
19 | 15, 17, 18 | 3syl 17 | . . . . . . . 8 |
20 | eleq1 2180 | . . . . . . . . . 10 | |
21 | 15, 20 | syl5ibcom 154 | . . . . . . . . 9 |
22 | 21 | orim2d 762 | . . . . . . . 8 |
23 | 19, 22 | mpd 13 | . . . . . . 7 |
24 | acexmidlem.b | . . . . . . . . . . . . . 14 | |
25 | pp0ex 4083 | . . . . . . . . . . . . . . 15 | |
26 | 25 | rabex 4042 | . . . . . . . . . . . . . 14 |
27 | 24, 26 | eqeltri 2190 | . . . . . . . . . . . . 13 |
28 | 27 | prid2 3600 | . . . . . . . . . . . 12 |
29 | 28, 6 | eleqtrri 2193 | . . . . . . . . . . 11 |
30 | eleq1 2180 | . . . . . . . . . . . . . . 15 | |
31 | 30 | anbi1d 460 | . . . . . . . . . . . . . 14 |
32 | 31 | rexbidv 2415 | . . . . . . . . . . . . 13 |
33 | 32 | reueqd 2613 | . . . . . . . . . . . 12 |
34 | 33 | rspcv 2759 | . . . . . . . . . . 11 |
35 | 29, 34 | ax-mp 5 | . . . . . . . . . 10 |
36 | riotacl 5712 | . . . . . . . . . 10 | |
37 | 35, 36 | syl 14 | . . . . . . . . 9 |
38 | elrabi 2810 | . . . . . . . . . 10 | |
39 | 38, 24 | eleq2s 2212 | . . . . . . . . 9 |
40 | elpri 3520 | . . . . . . . . 9 | |
41 | 37, 39, 40 | 3syl 17 | . . . . . . . 8 |
42 | eleq1 2180 | . . . . . . . . . 10 | |
43 | 37, 42 | syl5ibcom 154 | . . . . . . . . 9 |
44 | 43 | orim1d 761 | . . . . . . . 8 |
45 | 41, 44 | mpd 13 | . . . . . . 7 |
46 | 23, 45 | jca 304 | . . . . . 6 |
47 | anddi 795 | . . . . . 6 | |
48 | 46, 47 | sylib 121 | . . . . 5 |
49 | simpl 108 | . . . . . . 7 | |
50 | simpl 108 | . . . . . . 7 | |
51 | 49, 50 | jaoi 690 | . . . . . 6 |
52 | 51 | orim2i 735 | . . . . 5 |
53 | 48, 52 | syl 14 | . . . 4 |
54 | 53 | orcomd 703 | . . 3 |
55 | simpr 109 | . . . . 5 | |
56 | 55 | orim1i 734 | . . . 4 |
57 | 56 | orim2i 735 | . . 3 |
58 | 54, 57 | syl 14 | . 2 |
59 | 3orass 950 | . 2 | |
60 | 58, 59 | sylibr 133 | 1 |
Colors of variables: wff set class |
Syntax hints: wi 4 wa 103 wo 682 w3o 946 wceq 1316 wcel 1465 wral 2393 wrex 2394 wreu 2395 crab 2397 cvv 2660 c0 3333 csn 3497 cpr 3498 con0 4255 crio 5697 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-nul 4024 ax-pow 4068 |
This theorem depends on definitions: df-bi 116 df-3or 948 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-reu 2400 df-rab 2402 df-v 2662 df-sbc 2883 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-pw 3482 df-sn 3503 df-pr 3504 df-uni 3707 df-tr 3997 df-iord 4258 df-on 4260 df-suc 4263 df-iota 5058 df-riota 5698 |
This theorem is referenced by: acexmidlem1 5738 |
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